In this work, motivated by emerging low-latency applications, we consider an amplify-and-forward relaying network operating with finite blocklength (FBL) codes subject to delay quality of service (QoS) constraints. Hence, we address both transmission delay (via FBL codes) and queueing delay (via delay QoS requirements). We first derive the QoS-constrained throughput of the network. Subsequently, we state a resource allocation problem aiming at allocating the power between the source and the relay to maximize the throughput. The convexity of the problem is proved and the optimal power allocation policy is provided. Via simulations, we confirm the accurateness of our analytical model. In addition, we provide interesting insights on the system behavior by characterizing the impact of the error probability, the QoS-exponent and coding blocklength on the throughput performance.